The answer depends both on the parameter of interest and the experimental design.
The origins of the R and G variables do help in understanding for those not familiar with this technology. In 2-channel microarray experiments, you have 2 samples (e.g., collected under condition A and under condition B) of nucleic acid, with one sample labeled with a red (R) fluorescent marker and the other with green (G). The samples are mixed and applied to an array of spots, with each individual spot binding a specific nucleic-acid sequence. The ratio of R to G fluorescence on each spot then is used to measure the relative amounts of that particular nucleic-acid sequence in the 2 samples. The relative strengths of the R and G fluorescence signals on the array, however, could arise from different starting amounts of samples from A and B, different success in labeling the samples, differences in the volumes of the 2 labeled samples applied to the microarray, or different intrinsic intensities of fluorescent emission from the R and G markers.
From Wikipedia, the bias of an estimator is:
the difference between this estimator's expected value and the true value of the parameter being estimated.
So bias will mean something different if your parameter of interest is the fluorescence intensity per se, the ratio of amounts of nucleic acids in the labeled samples applied to the microarray, or the true ratio of amounts of some particular nucleic acid sequences between conditions A and B. I assume that you are interested in the latter.
If you do one experiment with sample A labeled with G and sample B labeled with R, the "normalization" tries to correct for the various technical ways in which G and R fluorescence might differ beyond the effects of the 2 conditions. The assumption is that most nucleic acid sequences don't differ in amount between the 2 conditions, so that a general correction for overall R/G signal ratio will correct for these potential technical difficulties. Only particular nucleic acids whose fluorescence ratios are adequately higher or lower than that overall R/G ratio would be considered significantly different between conditions A and B. So in terms of distinguishing the 2 conditions this is a correction for bias. In this case, with only one experiment, it's not very fruitful to talk about variance.
Now let's examine different labeling strategies for performing 6 replicates of this experiment. First, say that all samples from condition A are labeled G and all from condition B are labeled R. In that case a per-array normalization minimizes bias for each array (again, in terms of the ratio between conditions) and thus also the bias of an estimator based on the means of results among the 6 arrays. It's not immediately clear to me how much this would affect the variance of the estimator; my guess is probably little if at all.
Now say that instead you do 3 replicates with the above labeling scheme and 3 replicates with the labels reversed between the 2 conditions. In that case, bias in individual arrays in the log(A/B) ratio arising from general differences between R and G would average out even if there was no normalization. Even without normalization the true value of the log ratio between condition A and B, in this experimental design with balance of labels between conditions, could well be the expected value based on the ratios between the conditions (not colors) among the 6 arrays. In this case normalization does not affect bias; it does, however, greatly decrease variance among the 6 replicates.